This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: EECS 20. Solutions to Final Exam. December 16, 1999. 1. 20 points A signal is mathematically described as a function. As we have seen in the course, these may be functions of time and space or they may be sequences. For example, a onesecond long soundwave may be described as a function Sound : [0 , 1] → AirPressure . Propose mathematical descriptionss for the signals corresponding to the following intuitive descriptions. Give a brief justification for your answer. (a) 5 The signal obtained after sampling Sound (described above) 8,000 times. (b) 5 A blackandwhite 800 × 600 pixel image. (c) 5 A onesecond long blackandwhite video with 30 frames per second. (d) 5 The sequence of buttons you press with your TV remote control. Answer (a) We have SampledSound : { , 1 , ··· , 8 , 000 } → AirPressure , where for all n , SampledSound ( n ) = Sound ( n/ 8 , 000). (b) We have Image : { , ··· , 799 } × { , ··· , 599 } → { , ··· , 255 } , where the range is an 8bit colormap index. (c) We have Video : { , ··· , 29 } → Images , where Images are all functions of the form Image . (d) The sequence could be modeled as Ints → { 1 , ··· , 5 , open , close } , assuming that the available buttons are: 1 , ··· , 5 , open , close . 2. 20 points (a) 5 Write cos 3 ( ωt ) in terms of cosines and sines of multiples of ωt . (b) 5 Express in polar form all the distinct roots of the equation z 5 = 2. (c) 5 Find A and θ so that A sin( ωt + θ ) = sin( ωt ) + cos( ωt ) . (d) 5 Express the fraction 1+ jω 1 − jω in rectangular and polar forms. 1 x t1 1 1 Figure 1: The graph of x for problem 3 Answer (a) Write cos 3 ( ωt ) = 1 / 8[ e iωt + e − iωt ] 3 = 1 / 8[ e 3 iωt + e − 3 iωt + 3 e iωt + 3 e − ωt ] = 1 / 4[cos(3 ωt ) + 3cos( ωt ) . ] (b) The distinct roots are: { 2 1 / 5 × e i 2 πk/ 5  k = 0 , 1 , 2 , 3 , 4 } . (c) Since sin( ωt + θ ) = sin( ωt )cos( θ ) + cos( ωt )sin( θ ) = sin( ωt ) + cos( ωt ) , we must have A cos( θ ) = A sin( θ ) = 1 which gives A = √ 2 ,θ = π/ 4 . (d) We have 1 + jω 1 − jω = (1 + jω ) 2 1 + ω 2 = 1 − ω 2 1 + ω 2 + j 2 ω 1 + ω 2 , and 1 + jω 1 − jω = 1 × e j 2 tan 1 ω ....
View
Full
Document
This note was uploaded on 05/17/2009 for the course EE 20N taught by Professor Ayazifar during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Ayazifar
 Electrical Engineering

Click to edit the document details